Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each expression. Do not use a calculator. $$\log 10^{1.5}$$

Step-by-Step Solution

Answer

1.5

Step 1: Understanding the Problem

We need to evaluate the expression \( \log 10^{1.5} \). This involves finding the result of this logarithmic expression without using a calculator.

Step 2: Applying Logarithmic Identity

Recall the logarithmic identity \( \log_b (b^x) = x \). Here, the base \( b \) is 10, and the exponent \( x \) is 1.5. Therefore, by applying this identity, \( \log 10^{1.5} = 1.5 \).

Step 3: Final Evaluation

Using the identity, we conclude that \( \log 10^{1.5} \) simplifies directly to 1.5, as the base of the logarithm matches the base of the exponent.

Key Concepts

Logarithmic Identity

One of the fundamental concepts in logarithms is the logarithmic identity \( \log_b(b^x) = x \). This is an incredibly useful tool in mathematics because it simplifies the process of evaluating logarithmic expressions. Let's break it down to understand why this is the case.

• Logs are essentially the inverse operations of exponents. They "undo" what exponents "do."
• The expression \( b^x \) implies that the base \( b \) is multiplied by itself \( x \) number of times. For example, \( 10^{1.5} \) indicates the number 10 multiplied by itself 1.5 times.
• Applying this identity, when you take the logarithm of \( b^x \) with the same base \( b \), you "cancel out" the base, leaving you with the exponent \( x \).

Understanding this identity is crucial for simplifying logarithmic expressions quickly and efficiently. It allows students to focus on the properties of exponents themselves rather than the potentially tricky math of logarithms. Using this identity, solving our original problem becomes straightforward.

Logarithmic Evaluation

Logarithmic evaluation means solving expressions involving logarithms. Calculators help, but knowing how to evaluate them manually is important for building math skills and intuition. Evaluating a logarithm involves finding out what power you need to raise the base to get a certain number.

• In the expression \( \log 10^{1.5} \), you're trying to determine what power you need to raise 10 to get \( 10^{1.5} \).
• Using the logarithmic identity \( \log_b(b^x) = x \), it becomes simple. You "strip away" the logarithm and the base, leaving you with the exponent \( x \), which is 1.5.
• No need for complex arithmetic or calculator help—just apply the identity, and the job is done.

Mastering the evaluation of these expressions is a stepping stone to tackling more complex mathematical problems. Once you're comfortable with these basics, it becomes more instinctual to decompose and solve logarithmic problems without external aids.

Mathematics Education

Mathematics education involves teaching students not only how to solve problems but also to instill an understanding of mathematical concepts. Logarithms are integral to many fields of study, such as science and engineering, and learning them can greatly benefit real-world problem solving.

• Utilizing logarithmic identities simplifies problem-solving, providing a clear understanding of powering and inverse operations.
• Effective teaching should focus on practical applications, encouraging students to explore how these identities work in tangible scenarios.
• By understanding that \( \log_b(b^x) = x \), students can approach logarithmic problems with the confidence that comes from knowing the "why" behind the "how."

Encouraging a deep understanding of concepts rather than rote memorization empowers students in their educational journey. This concept-focused approach not only applies to logarithms but to mathematics education as a whole, fostering critical thinking and analytical skills.